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Two Simple Models for Observational Studies

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Design of Observational Studies

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Abstract

Observational studies differ from experiments in that randomization is not used to assign treatments. How were treatments assigned? This chapter introduces two simple models for treatment assignment in observational studies. The first model is useful but naïve: it says that people who look comparable are comparable. The second model speaks to a central concern in observational studies: people who look comparable in the observed data may not actually be comparable; they may differ in ways we did not observe.

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Notes

  1. 1.

    The alert reader will notice that the supposition that there could be two subjects, k and ℓ, with Z k + Z ℓ = 1 but π k = π ℓ is already a step beyond the mere representation in (3.1), because it precludes setting u i = Z i for all i and instead requires 0 < π i < 1 for at least two subjects.

  2. 2.

    As in Sect. 1.2.6, this definition of Z k leaves open the possibility of using cohort size Z k as an instrument for the actual size of the class, and indeed Angrist and Lavy [5] reported instrumental variable analyses.

  3. 3.

    The argument just presented is the simplest version of a general argument developed in [81].

  4. 4.

    See, for instance, the argument made by David Gross and Nicholas Souleles, quoted in Sect. 4.2 from their paper [40]. The example in Chap. 8 attempts to use both approaches, although it is, perhaps, not entirely convincing in either attempt. In Chap. 8 , the dose of chemotherapy received by patients with ovarian cancer is varied by comparing two types of providers of chemotherapy, gynecological oncologists (GO) and medical oncologists (MO), where the latter treat more intensively. The most important clinical covariates are measured, including clinical stage, tumor grade, histology, and comorbid conditions. Despite these two good attempts, it is far from clear that the GO-vs-MO assignment is haphazard, even after adjustments, and it is quite likely that the oncologist knew considerably more about the patient than is recorded in electronic form.

  5. 5.

    Figure 1.2 is an example of quasi-experimental reasoning. Instead of comparing college attendance in two groups, say the children of deceased fathers in 1979–1981, when the benefit program was available, and the children of deceased fathers in 1982–1983, after the program was withdrawn, Fig. 1.2 follows Susan Dynarski’s [26] study in comparing four groups. Figure 1.2 shows at most a small shift in college attendance by the group of children without deceased fathers, who were never eligible for the program. This is a small check, but a useful one: it suggests that the comparison of children of deceased fathers in 1979–1981 and 1982–1983, though totally confounded with time, may not be seriously biased by this confounding, because a substantial shift in college attendance by unaffected groups over this time is not evident. The quasi-experimental design in Fig. 1.2 is widely used and is known by a variety of not entirely helpful names, including “pretest–posttest nonequivalent control group design” and “difference-in-differences design.” Kevin Volpp et al. [131] used this design in a medical context to study the possible effects on patient mortality of a reduction, to 80 h per week, of the maximum number of hours that a medical resident may work in a hospital. See [1, 7, 9, 14, 92] for discussion in general terms. The study by Garen Wintemute et al. [135], as discussed in Sect. 4.3, contains another example of quasi-experimental reasoning using a “control construct,” that is, an outcome thought to be unaffected by the treatment. They were interested in the possible effects of a change in gun control laws on violent crimes using guns. One would not expect a change in gun control laws to have a substantial effect on nonviolent, nongun crimes, and so the rate of such crimes is their control construct or unaffected outcome. See Sect. 4.3. An example with two control groups subject to different biases is discussed in Sect. 12.3.

  6. 6.

    Technically, strongly ignorable treatment assignment given x is \( \left . Z\, \underline {\;\left \vert \;\right \vert \;}\,\left ( r_{T},\,r_{C}\right ) \,\right \vert \,\mathbf {x}\) and \(0<\Pr \left ( \left . Z=1\,\right \vert \,\, \mathbf {x}\right ) <1\) for all x, or in a single expression, \( 0<\Pr \left ( \left . Z=1\,\right \vert \,r_{T},\,r_{C},\,\mathbf {x}\right ) \) \( =\Pr \left ( \left . Z=1\,\right \vert \,\,\mathbf {x}\right ) <1\) for all x; that is, no explicit reference is made to an unobserved covariate u. I introduce u here, rather than in Sect. 3.4 where it is essential, to simplify the transition between Sects. 3.3 and 3.4.

    The remainder of this footnote is a slightly technical discussion of the formal relationship between (3.5) and the condition as given in [104]; it will interest only the insanely curious. In a straightforward way [23, Lemma 3] , condition (3.5) implies \(\left . Z\, \underline { \;\left \vert \;\right \vert \;}\,\left ( r_{T},\,r_{C}\right ) \,\right \vert \, \mathbf {x}\) and \(0<\Pr \left ( \left . Z=1\,\right \vert \,\,\mathbf {x}\right ) <1\) for all x. Also, in a straightforward way [23, Lemma 3] , if (3.5) is true then (i) \(\left . Z\, \underline { \;\left \vert \;\right \vert \;}\,\left ( r_{T},\,r_{C}\right ) \,\right \vert \,\left ( \mathbf {x},u\right ) \), (ii) \(0<\Pr \left ( \left . Z=1\,\right \vert \,\,\mathbf {x}\right ) <1\) for all x, and (iii) \(\left . Z\, \underline {\;\left \vert \;\right \vert \;}u\,\right \vert \,\mathbf {x}\) are true. Moreover, (ii) and (iii) together imply (iv) \(0<\Pr \left ( \left . Z=1\,\right \vert \,\,\mathbf {x},u\right ) <1\) for all \(\left ( \mathbf {x} ,u\right ) \). In words, condition (3.5) implies both strong ignorability given x and also strong ignorability given \(\left ( \mathbf {x},u\right ) \). Now, conditions (i) and (iv), without condition (iii), are the key elements of the sensitivity model in Sect. 3.4: they say that treatment assignment would have been strongly ignorable if u had been measured and included with x; so the failure to measure u is the source of our problems. Moreover, if condition (iii) were true in addition to (i) and (iv), then (3.5) and strong ignorability given x follow. In brief, strong ignorability given x is implied by the addition of condition (iii) to strong ignorability given \(\left ( \mathbf {x} ,u\right ) \). In words, one can reasonably think of the naive model Sect. 3.3 as the sensitivity model of Sect. 3.4 together with the irrelevance of u as expressed by \(\left . Z\, \underline {\;\left \vert \;\right \vert \;}u\,\right \vert \, \mathbf {x}\) in condition (iii).

  7. 7.

    The proof is easy [104]. Recall the following basic facts about conditional expectations: if A, B, and C are random variables, then \(E\left ( A\right ) =E\left \{ E\left ( \left . A \mathbf {\,}\right \vert \,\,B\,\right ) \right \} \) and \(E\left ( \left . A \mathbf {\,}\right \vert \,\,C\,\right ) =E\left \{ \left . E\left ( \left . A \mathbf {\,}\right \vert \,\,B,C\,\right ) \,\right \vert \,C\right \} \). From the definition of conditional independence, to prove (3.10), it suffices to prove \(\Pr \left \{ \left . Z=1 \mathbf {\,}\right \vert \,\,\mathbf {x},\,e\left ( \mathbf {x}\right ) \right \} =\Pr \left \{ \left . Z=1\mathbf {\,}\right \vert \,\,\,e\left ( \mathbf {x} \right ) \right \} \). Because \(e\left ( \mathbf {x}\right ) \) is a function of x, conditioning on x fixes \(e\left ( \mathbf {x}\right ) \) , so \(\Pr \left \{ \left . Z=1\mathbf {\,}\right \vert \,\,\mathbf {x},\,e\left ( \mathbf {x}\right ) \right \} \) \(=\Pr \left ( \left . Z=1\mathbf {\,}\right \vert \,\,\mathbf {x}\right ) =e\left ( \mathbf {x}\right ) \). Because Z = 0 or Z = 1 , for any random variable D, we have \(\Pr \left ( \left . Z=1\mathbf {\,} \right \vert \,\,D\right ) =E\left ( \left . Z\mathbf {\,}\right \vert \,D\right ) \) . With A = Z, B = x, \(C=e\left ( \mathbf {x}\right ) \), we have \(\Pr \left \{ \left . Z=1\mathbf {\,}\right \vert \,\,\,e\left ( \mathbf {x}\right ) \right \} =E\left \{ \left . Z\mathbf {\,}\right \vert \,\,e\left ( \mathbf {x} \right ) \,\right \} \) \(=E\left \{ \left . E\left \{ \left . Z\mathbf {\,} \right \vert \,\,\mathbf {x},\,e\left ( \mathbf {x}\right ) \right \} \,\right \vert \,e\left ( \mathbf {x}\right ) \right \} \) \(=E\left \{ \left . \Pr \left \{ \left . Z=1\mathbf {\,}\right \vert \,\,\mathbf {x},\,e\left ( \mathbf {x} \right ) \right \} \,\right \vert \,e\left ( \mathbf {x}\right ) \right \} \) \( =E\left \{ \left . e\left ( \mathbf {x}\right ) \,\,\right \vert \,e\left ( \mathbf { x}\right ) \right \} \) \(=e\left ( \mathbf {x}\right ) =\Pr \left ( \left . Z=1 \mathbf {\,}\right \vert \,\,\mathbf {x}\right ) \), which is what we needed to prove.

  8. 8.

    Part II makes use of a stronger version of the balancing property than stated in (3.10). Specifically, if you match on \( e\left ( \mathbf {x}\right ) \) and any other aspect of x, say \( h\left ( \mathbf {x}\right ) \), you still balance x, that is, \( \left . Z\, \underline {\;\left \vert \;\right \vert \;}\,\mathbf {x}\,\right \vert \,\left \{ e\left ( \mathbf {x}\right ) ,\,h\left ( \mathbf {x}\right ) \right \} \) for any \(h\left ( \mathbf {x}\right ) \). See [104] for the small adjustments required in the proof.

  9. 9.

    Theoretical arguments also show that estimated propensity scores can have better properties than true propensity scores; see [81, 85] .

  10. 10.

    In technical terms, if treatment assignment is strongly ignorable given x, then it is strongly ignorable given \(e\left ( \mathbf {x}\right ) \) . The proof of this version is also short [104]. In footnote 7, we saw that \(\Pr \left \{ \left . Z=1\mathbf {\, }\right \vert \,\,\,e\left ( \mathbf {x}\right ) \right \} =\Pr \left ( \left . Z=1 \mathbf {\,}\right \vert \,\,\mathbf {x}\right ) \). Condition (3.11) says \(\Pr \left ( \left . Z=1\,\right \vert \,r_{T},\,r_{C},\,\mathbf {x},\,u\right ) =\Pr \left ( \left . Z=1\,\right \vert \,\,\mathbf {x}\right ) \). Together they say \(\Pr \left ( \left . Z=1\,\right \vert \,r_{T},\,r_{C},\,\mathbf {x},\,u\right ) =\Pr \left \{ \left . Z=1\mathbf {\,}\right \vert \,\,\,e\left ( \mathbf {x}\right ) \right \} \), which implies (3.12).

  11. 11.

    In general, a sensitivity analysis asks how the conclusion of an argument dependent upon assumptions would change if the assumptions were relaxed. The term is sometimes misused to refer to performing several parallel statistical analyses without regard to the assumptions upon which they depend. If several statistical analyses all depend upon the same assumption—for instance, the naïve model (3.5)—then performing several such analyses provides no insight into consequences of the failure of that assumption.

  12. 12.

    Odds are an alternative way of expressing probabilities. Probabilities and odds carry the same information in different forms. A probability of π k = 2∕3 is an odds of \(\pi _{k}/\left ( 1-\pi _{k}\right ) =2\) or 2-to-1. Gamblers prefer odds to probabilities because odds express the chance of an event in terms of fair betting odds, the price of a fair bet. It is easy to move from probability π k to odds \(\omega _{k}=\pi _{k}/\left ( 1-\pi _{k}\right ) \) and back again from odds ω k to probability \(\pi _{k}=\omega _{k}/\left ( 1+\omega _{k}\right ) \).

  13. 13.

    Implicitly, the critic is saying that the failure to measure u is the source of the problem, or that (3.5) would be true with \(\left ( \mathbf {x} ,u\right ) \) in place of x, but is untrue with x alone. That is, the critic is saying \(\pi _{\ell }=\Pr \left ( \left . Z_{\ell }=1\,\right \vert \,r_{T\ell },\,r_{C\ell },\,{\mathbf {x}}_{\ell },\,u_{\ell }\right ) \) \(=\Pr \left ( \left . Z_{\ell }=1\,\right \vert \,{\mathbf {x}}_{\ell },\,u_{\ell }\right ) \). As in Sect. 3.1, because of the delicate nature of unobserved variables, this is a manner of speaking rather than a tangible distinction. If the formalities are understood to refer to \(\pi _{\ell }=\Pr \left ( \left . Z_{\ell }=1\,\right \vert \,r_{T\ell },\,r_{C\ell },\,{\mathbf {x}}_{\ell },\,u_{\ell }\right ) \), then it is not necessary to insist that \(\pi _{\ell }=\Pr \left ( \left . Z_{\ell }=1\,\right \vert \,\,{\mathbf {x}}_{\ell },\,u_{\ell }\right ) \). Conversely, there is always a scalar unobserved covariate u with 0 ≤ u ≤ 1 such that \(\left . Z\, \underline { \;\left \vert \;\right \vert \;}\,\left ( r_{T},\,r_{C}\right ) \,\right \vert \,\left ( \mathbf {x},u\right ) \), namely the unobserved covariate u = ζ, where \(\zeta =\Pr \left ( Z=1 | r_{T}, r_{C}, \mathbf {x} \right )\); for proof, see [102, §6.3]. Consistent with terminology introduced by Frangakis and Rubin [33], we might call this unobserved covariate, u = ζ, the “principal unobserved covariate.” In effect, the bounded, scalar principal unobserved covariate, u = ζ, is the only unobserved covariate that matters, and it always exists. A finite Γ changes the principal unobserved covariate, u = ζ, from a representation to a model, because a finite Γ implies 0 < ζ < 1 rather than 0 ≤ ζ ≤ 1.

  14. 14.

    In a fussy technical sense, the numbering of pairs and people within pairs is supposed to convey nothing about these people, except that they were eligible to be paired, that is, they have the same observed covariates, different treatments, with 2I distinct people. Information about people is supposed to be recorded in variables that describe them, such as Z, x, u, r T, r C, not in their position in the data set. You cannot put your brother-in-law in the last pair just because of that remark he made last Thanksgiving; you have to code him in an explicit brother-in-law variable. Obviously, it is easy to make up subscripts that meet this fussy requirement: number the pairs at random, then number the people in a pair at random. The fussy technical point is that, in going from the L people in (3.1) to the 2I paired people, no information has been added and tucked away into the subject numbers—the criteria for pairs are precisely x i1 = x i2 , Z i1 + Z i2 = 1 with 2I distinct individuals.

  15. 15.

    What does it mean to speak of the “largest distribution”? One random variable, A, is said to be stochastically larger than another random variable, B, if \(\Pr \left ( A\geq k\right ) \geq \Pr \left ( B\geq k\right ) \) for every k. That is, A is more likely than B to jump over a bar at height k, no matter how high, k, the bar is set. Because this must be true for every k, it is a rather special relationship between random variables. For instance, it might happen that \(\Pr \left ( A\geq -1.65\right ) =0.95\), \(\Pr \left ( A\geq 1.65\right ) =0.05\) while \(\Pr \left ( B\geq -1.65\right ) =0.80\), \(\ \Pr \left ( B\geq 1.65\right ) =0.20\), so neither A nor B is stochastically larger than the other. For instance, this is true if A is Normal with mean zero and standard deviation 1, and B is Normal with mean zero and standard deviation 2. The intuition is overwhelmingly strong that the largest null distribution of Wilcoxon’s signed rank statistic is obtained by driving the chance of a positive difference up to its maximum, namely \(\varGamma /\left ( 1+\varGamma \right ) \) in (3.18), and this intuition turns out to be correct. It takes a small amount of effort in this case to show the distribution is actually stochastically largest; see [93, Chapter 4] . In other cases, the same result is available with somewhat more effort. In still other cases, one can speak of a “largest distribution” only asymptotically, that is, only in large samples; see [37, 101] or [93, Chapter 4]. This presents no problem in practice, because the asymptotic results are quite adequate and easy to use [93, Chapter 4] ; however, it does make the theory of the paired case simpler than, say, the theory of matching with several controls matched to each treated subject. To see a parallel discussion of the paired case and the case of several controls, see [97]. For technical details, see [101].

  16. 16.

    See [88] or [93, Chapter 4] .

  17. 17.

    In contrast, for a Normal distribution with positive expectation, abz(w) → 1 as w →∞, so (3.29) does not hold for any Δ < ∞. Looking at Figure 3 of [98, §3], this fact seems to be a peculiarity of the Normal distribution with its exceptionally thin tails, not a peculiarity of (3.29).

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Appendix: Exact Computations for Sensitivity Analysis

Appendix: Exact Computations for Sensitivity Analysis

For small to moderate I, the exact upper bound on the distribution of Wilcoxon’s signed rank statistic may be obtained quickly in R as the convolution of I probability generating functions. Only the situation without ties is considered. Pagano and Tritchler [73] observed that permutation distributions may often be obtained in polynomial time by applying the fast Fourier transform to the convolution of characteristic functions or generating functions. Here, probability generating functions are used along with the R function convolve.

The distribution of \(\overline {\overline {T}}\) is the distribution of the sum of I independent random variables, i = 1, 2, …, I, taking the value i with probability \(\varGamma /\left ( 1+\varGamma \right ) \) and the value 0 with probability \(1/\left ( 1+\varGamma \right ) \). The ith random variable has probability generating function

$$\displaystyle \begin{aligned} h_{i}\left( x\right) =\frac{1}{1+\varGamma }+\frac{\varGamma x^{i}}{1+\varGamma } , \end{aligned} $$
(3.30)

and \(\overline {\overline {T}}\) has generating function \(\varPi _{i=1}^{I}h_{i}\left ( x\right ) \). In R, the generating function of a random variable taking integer values 0, 1, …, B is represented by a vector of dimension B + 1 whose b + 1 coordinate gives the probability that the random variable equals b. For instance, \(h_{3}\left ( x\right ) \) is represented by

$$\displaystyle \begin{aligned} \left( \frac{1}{1+\varGamma },\;0,\;0,\;\frac{\varGamma }{1+\varGamma }\right) \text{. } \end{aligned} $$
(3.31)

The distribution of \(\overline {\overline {T}}\) is obtained by convolution as a vector with \(1+I\left ( I+1\right ) /2\) coordinates representing \(\varPi _{i=1}^{I}h_{i}\left ( x\right ) \) and giving \(\Pr \left ( \overline { \overline {T}}=b\right ) \) for b = 0, 1, …, \(I\left ( I+1\right ) /2\), where \(I\left ( I+1\right ) /2=\sum i\).

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R. Rosenbaum, P. (2020). Two Simple Models for Observational Studies. In: Design of Observational Studies. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-46405-9_3

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